The Stefan problem with small surface tension

Abstract:

The Stefan problem with small surface tension $\varepsilon$ is considered. Assuming that the classical Stefan problem (with $\varepsilon = 0$) has a smooth free boundary $\Gamma$, we denote the temperature of the solution by ${\theta _0}$ and consider an approximate solution ${\theta _0} + \varepsilon u$ for the case where $\varepsilon \ne 0$, $\varepsilon$ small. We first establish the existence and uniqueness of $u$, and then investigate the effect of $u$ on the free boundary $\Gamma$. It is shown that small surface tension affects the free boundary $\Gamma$ radically differently in the two-phase problem than in the one-phase problem.